GENERALIZED TRIGONOMETRIC SYSTEMS AND SPECTRAL TASKS WITH ADDITIONAL INTERNAL BOUNDARY CONDITIONS
Abstract
When solving problems related to the study of the strength properties of various structures, some sets of trigonometric (sine or cosine), as well as hyperbolic functions are often used, which cyclically pass into each other when taking derivatives. These sets consist of two functions, and the last of these functions, when differentiating, passes into the first, taken respectively with a plus sign (a trigonometric system of the first type) or a minus sign (a trigonometric system of the second type). Trigonometric and hyperbolic functions are also used in solving many applied problems, whose mathematical models contain second derivatives in spatial variables. If the mathematical model contains fourth-order derivatives with respect to spatial variables, then when solving the corresponding problems, it is possible to use functions whose fourth derivatives are proportional to these functions. There are a number of works on the general theory of systems of functions, where generalized trigonometric systems (GTS) of functions are described, the derivatives of a certain order of which are proportional to these functions. In this paper, this theory is developed in the direction of studying the quadratic forms of the functions that make up the GTS. It is shown that the quadratic forms of GTS functions can themselves be GTS functions of the same order (of the first or second types). The obtained identities and the created theory are used to solve spectral problems for a fourth-order operator for functions with certain conditions. The specificity of the problems under consideration is that in addition to the standard boundary conditions, there are additional conditions on the inner boundary. These conditions are not sufficient to independently solve the problem in each separate domain in which the functions under study are specified. The use of the GTS properties identified herein allows us to solve such problems in the entire area under consideration.
